Lesson 7: 5.7 Using the Second Derivative Test
Duration of Days: 2
Lesson Objective
Identify Critical Points: Find values of c where f'(c) = 0 (Note: the Second Derivative Test is specifically used for locations where the derivative is zero, not necessarily where it is undefined).
Determine Concavity at a Point: Evaluate f''(c) to determine if the graph is concave up or concave down at the critical point.
Classify Extrema: Conclude that a relative minimum exists if f'(c) = 0 and f''(c) > 0.
Conclude that a relative maximum exists if f'(c) = 0 and f''(c) < 0.
Recognize Inconclusive Results: Identify when f''(c) = 0 or is undefined, necessitating a return to the First Derivative Test.
Communicate Justification: Use formal AP-style phrasing: "Since f'(c) = 0 and f''(c) < 0, f has a relative maximum at x = c."
How can the concavity of a function at a critical point determine if that point is a relative maximum or minimum?
What are the specific conditions under which the Second Derivative Test fails, and what is the "plan B" when it does?
How does the Second Derivative Test differ in application and purpose from the First Derivative Test?
Critical Point (specifically where f'(x) = 0)
Relative (Local) Extrema
Relative Maximum
Relative Minimum
Concavity
Concave Up
Concave Down
Second Derivative
Point of Inflection
Inconclusive Test
First Derivative Test (for comparison)
Twice-Differentiable Function
FUN-4.A: Justify conclusions about the behavior of a function based on the sign of its derivatives.
FUN-4.A.1: The second derivative test can be used to identify a relative maximum or a relative minimum of a function.
This lesson introduces the Second Derivative Test as an alternative method for identifying local extrema. Students will learn that if f'(c) = 0 (a horizontal tangent) and f''(c) exists, the sign of the second derivative tells us the "shape" of the graph at that point, thereby confirming the type of extremum.
Purpose: To provide a more efficient way to classify extrema when the second derivative is easy to compute (such as with polynomials) or when only point data (not interval data) is provided. This is a common requirement in AP Free Response questions where a table of values is given.
DOK Level 2 (Skill/Concept): Applying the test to polynomial or simple transcendental functions to find relative extrema.
DOK Level 3 (Strategic Thinking): Determining which test (First or Second Derivative) is most appropriate for a given function or set of data. Justifying extrema using the specific formal language required by the AP College Board.
The "Smiley/Frowny" Mnemonic: Use visual cues. A positive second derivative makes a smile (minimum); a negative second derivative makes a frown (maximum).
Step-by-Step Checklist:
Create a flow chart: Find f'(x).
Set to zero to find c.
Find f''(x).
Plug c into f''(x).
Interpret the sign.
Comparison Table: Provide a side-by-side comparison of the First vs. Second Derivative Test to show that both lead to the same conclusion but use different "evidence."
For Advanced Learners (Extension):Inconclusive Investigations: Ask students to explain why the Second Derivative Test fails and what that implies about the point (it's not necessarily an inflection point).
Implicit Differentiation: Challenge students to use the Second Derivative Test on a curve defined implicitly, such as a circle or hyperbola, where finding y'' requires substituted algebra.The "Why" of Failure: Explore why the test fails at a point where the second derivative is zero, linking it back to the definition of concavity and Taylor Polynomials.
AP College Board Assessment